QBManager

Rotational Motion

149716 If the external forces acting on a system have zero resultant, the centre of mass:

1 may move but not accelerate

2 may accelerate

3 must not move

4 None of the above

Explanation:

A Motion of the centre of mass,
$Ma {\vec{a}}_{CM} = {\vec{F}}_{ext}$
$F_{ext} = 0, {\vec{a}}_{CM} = 0$
${\vec{v}}_{CM} = constant$
If no external force act on a system the velocity of its centre of mass remains constant. Thus, the centre of mass move but not accelerate.

UPSEE - 2005

Rotational Motion

149723 A ladder leaned against a smooth wall and it is allowed to slip on a frictionless floor. Which figure represents the track of its centre of mass?

1

2

3

4

Explanation:

A
There are force on ladder $Mg$ normal force from the floor and normal force from the wall.
So, the Net force will be in -ve due to $Mg - N$ and $- ve$ due to the normal from the wall.
Finally the rod will come to rest on the ground so we can see that it has moved down and in left direction.

AIIMS-2005

Rotational Motion

149735 Two particles of mass $1 kg$ and $3 kg$ have position vectors $(2 \hat{i} + 3 \hat{j} + 4 \hat{k})$ and $(- 2 \hat{i} + 3 \hat{j} - 4 \hat{k})$ respectively. The centre of mass, has a position vector

1

\hat{i} + 3 \hat{j} - 2 \hat{k}

2

- \hat{i} - 3 \hat{j} - 2 \hat{k}

3

- \hat{i} + 3 \hat{j} + 2 \hat{k}

4

- \hat{i} + 3 \hat{j} - 2 \hat{k}

Explanation:

D Let $m_{1} = 1 kg, m_{2} = 3 kg$
${\vec{r}}_{1} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} and {\vec{r}}_{2} = - 2 \hat{i} + 3 \hat{j} - 4 \hat{k}$
$∴ {\vec{r}}_{CM} = \frac{m_{1} {\vec{r}}_{1} + m_{2} {\vec{r}}_{2}}{m_{1} + m_{2}}$
$= \frac{1 (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) + 3 (- 2 \hat{i} + 3 \hat{j} - 4 \hat{k})}{1 + 3}$
${\vec{r}}_{CM} = - \hat{i} + 3 \hat{j} - 2 \hat{k}$

EAMCET-1999

Rotational Motion

149736 The velocities of three particles of masses $20 g$ , $30 g$ and $50 g$ are $10 \hat{i}, 10 \hat{j}$ and $10 \hat{k}$ respectively. The velocity of the centre of mass of the three particles is

1

2 \hat{i} + 3 \hat{j} + 5 \hat{k}

2

10 (\hat{i} + \hat{j} + \hat{k})

3

20 \hat{i} + 30 \hat{j} + 5 \hat{k}

4

2 \hat{i} + 30 \hat{j} + 50 \hat{k}

Explanation:

A Let $m_{1}, m_{2}, m_{3}$ be $20 g, 30 g, 50 g$ respectively and ${\vec{V}}_{1}, {\vec{V}}_{2}, {\vec{V}}_{3}$ be $10 \hat{i}, 10 \hat{j}, 10 \hat{k}$ respectively
Then, ${\vec{V}}_{CM} = \frac{m_{1} {\vec{V}}_{1} + m_{2} {\vec{V}}_{2} + m_{3} {\vec{V}}_{3}}{m_{1} + m_{2} + m_{3}}$
$= \frac{20 \times 10 \hat{i} + 30 \times 10 \hat{j} + 50 \times 10 \hat{k}}{20 + 30 + 50}$
$= \frac{200 \hat{i} + 300 \hat{j} + 500 \hat{k}}{100}$
${\vec{V}}_{C M} = 2 \hat{i} + 3 \hat{j} + 5 \hat{k}$

EAMCET-2001

Rotational Motion

149716 If the external forces acting on a system have zero resultant, the centre of mass:

1 may move but not accelerate

2 may accelerate

3 must not move

4 None of the above

Explanation:

A Motion of the centre of mass,
$Ma {\vec{a}}_{CM} = {\vec{F}}_{ext}$
$F_{ext} = 0, {\vec{a}}_{CM} = 0$
${\vec{v}}_{CM} = constant$
If no external force act on a system the velocity of its centre of mass remains constant. Thus, the centre of mass move but not accelerate.

UPSEE - 2005

Rotational Motion

149723 A ladder leaned against a smooth wall and it is allowed to slip on a frictionless floor. Which figure represents the track of its centre of mass?

1

2

3

4

Explanation:

A
There are force on ladder $Mg$ normal force from the floor and normal force from the wall.
So, the Net force will be in -ve due to $Mg - N$ and $- ve$ due to the normal from the wall.
Finally the rod will come to rest on the ground so we can see that it has moved down and in left direction.

AIIMS-2005

Rotational Motion

149735 Two particles of mass $1 kg$ and $3 kg$ have position vectors $(2 \hat{i} + 3 \hat{j} + 4 \hat{k})$ and $(- 2 \hat{i} + 3 \hat{j} - 4 \hat{k})$ respectively. The centre of mass, has a position vector

1

\hat{i} + 3 \hat{j} - 2 \hat{k}

2

- \hat{i} - 3 \hat{j} - 2 \hat{k}

3

- \hat{i} + 3 \hat{j} + 2 \hat{k}

4

- \hat{i} + 3 \hat{j} - 2 \hat{k}

Explanation:

D Let $m_{1} = 1 kg, m_{2} = 3 kg$
${\vec{r}}_{1} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} and {\vec{r}}_{2} = - 2 \hat{i} + 3 \hat{j} - 4 \hat{k}$
$∴ {\vec{r}}_{CM} = \frac{m_{1} {\vec{r}}_{1} + m_{2} {\vec{r}}_{2}}{m_{1} + m_{2}}$
$= \frac{1 (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) + 3 (- 2 \hat{i} + 3 \hat{j} - 4 \hat{k})}{1 + 3}$
${\vec{r}}_{CM} = - \hat{i} + 3 \hat{j} - 2 \hat{k}$

EAMCET-1999

Rotational Motion

149736 The velocities of three particles of masses $20 g$ , $30 g$ and $50 g$ are $10 \hat{i}, 10 \hat{j}$ and $10 \hat{k}$ respectively. The velocity of the centre of mass of the three particles is

1

2 \hat{i} + 3 \hat{j} + 5 \hat{k}

2

10 (\hat{i} + \hat{j} + \hat{k})

3

20 \hat{i} + 30 \hat{j} + 5 \hat{k}

4

2 \hat{i} + 30 \hat{j} + 50 \hat{k}

Explanation:

A Let $m_{1}, m_{2}, m_{3}$ be $20 g, 30 g, 50 g$ respectively and ${\vec{V}}_{1}, {\vec{V}}_{2}, {\vec{V}}_{3}$ be $10 \hat{i}, 10 \hat{j}, 10 \hat{k}$ respectively
Then, ${\vec{V}}_{CM} = \frac{m_{1} {\vec{V}}_{1} + m_{2} {\vec{V}}_{2} + m_{3} {\vec{V}}_{3}}{m_{1} + m_{2} + m_{3}}$
$= \frac{20 \times 10 \hat{i} + 30 \times 10 \hat{j} + 50 \times 10 \hat{k}}{20 + 30 + 50}$
$= \frac{200 \hat{i} + 300 \hat{j} + 500 \hat{k}}{100}$
${\vec{V}}_{C M} = 2 \hat{i} + 3 \hat{j} + 5 \hat{k}$

EAMCET-2001

Rotational Motion

149716 If the external forces acting on a system have zero resultant, the centre of mass:

1 may move but not accelerate

2 may accelerate

3 must not move

4 None of the above

Explanation:

A Motion of the centre of mass,
$Ma {\vec{a}}_{CM} = {\vec{F}}_{ext}$
$F_{ext} = 0, {\vec{a}}_{CM} = 0$
${\vec{v}}_{CM} = constant$
If no external force act on a system the velocity of its centre of mass remains constant. Thus, the centre of mass move but not accelerate.

UPSEE - 2005

Rotational Motion

149723 A ladder leaned against a smooth wall and it is allowed to slip on a frictionless floor. Which figure represents the track of its centre of mass?

1

2

3

4

Explanation:

A
There are force on ladder $Mg$ normal force from the floor and normal force from the wall.
So, the Net force will be in -ve due to $Mg - N$ and $- ve$ due to the normal from the wall.
Finally the rod will come to rest on the ground so we can see that it has moved down and in left direction.

AIIMS-2005

Rotational Motion

149735 Two particles of mass $1 kg$ and $3 kg$ have position vectors $(2 \hat{i} + 3 \hat{j} + 4 \hat{k})$ and $(- 2 \hat{i} + 3 \hat{j} - 4 \hat{k})$ respectively. The centre of mass, has a position vector

1

\hat{i} + 3 \hat{j} - 2 \hat{k}

2

- \hat{i} - 3 \hat{j} - 2 \hat{k}

3

- \hat{i} + 3 \hat{j} + 2 \hat{k}

4

- \hat{i} + 3 \hat{j} - 2 \hat{k}

Explanation:

D Let $m_{1} = 1 kg, m_{2} = 3 kg$
${\vec{r}}_{1} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} and {\vec{r}}_{2} = - 2 \hat{i} + 3 \hat{j} - 4 \hat{k}$
$∴ {\vec{r}}_{CM} = \frac{m_{1} {\vec{r}}_{1} + m_{2} {\vec{r}}_{2}}{m_{1} + m_{2}}$
$= \frac{1 (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) + 3 (- 2 \hat{i} + 3 \hat{j} - 4 \hat{k})}{1 + 3}$
${\vec{r}}_{CM} = - \hat{i} + 3 \hat{j} - 2 \hat{k}$

EAMCET-1999

Rotational Motion

149736 The velocities of three particles of masses $20 g$ , $30 g$ and $50 g$ are $10 \hat{i}, 10 \hat{j}$ and $10 \hat{k}$ respectively. The velocity of the centre of mass of the three particles is

1

2 \hat{i} + 3 \hat{j} + 5 \hat{k}

2

10 (\hat{i} + \hat{j} + \hat{k})

3

20 \hat{i} + 30 \hat{j} + 5 \hat{k}

4

2 \hat{i} + 30 \hat{j} + 50 \hat{k}

Explanation:

A Let $m_{1}, m_{2}, m_{3}$ be $20 g, 30 g, 50 g$ respectively and ${\vec{V}}_{1}, {\vec{V}}_{2}, {\vec{V}}_{3}$ be $10 \hat{i}, 10 \hat{j}, 10 \hat{k}$ respectively
Then, ${\vec{V}}_{CM} = \frac{m_{1} {\vec{V}}_{1} + m_{2} {\vec{V}}_{2} + m_{3} {\vec{V}}_{3}}{m_{1} + m_{2} + m_{3}}$
$= \frac{20 \times 10 \hat{i} + 30 \times 10 \hat{j} + 50 \times 10 \hat{k}}{20 + 30 + 50}$
$= \frac{200 \hat{i} + 300 \hat{j} + 500 \hat{k}}{100}$
${\vec{V}}_{C M} = 2 \hat{i} + 3 \hat{j} + 5 \hat{k}$

EAMCET-2001

Rotational Motion

149716 If the external forces acting on a system have zero resultant, the centre of mass:

1 may move but not accelerate

2 may accelerate

3 must not move

4 None of the above

Explanation:

A Motion of the centre of mass,
$Ma {\vec{a}}_{CM} = {\vec{F}}_{ext}$
$F_{ext} = 0, {\vec{a}}_{CM} = 0$
${\vec{v}}_{CM} = constant$
If no external force act on a system the velocity of its centre of mass remains constant. Thus, the centre of mass move but not accelerate.

UPSEE - 2005

Rotational Motion

149723 A ladder leaned against a smooth wall and it is allowed to slip on a frictionless floor. Which figure represents the track of its centre of mass?

1

2

3

4

Explanation:

A
There are force on ladder $Mg$ normal force from the floor and normal force from the wall.
So, the Net force will be in -ve due to $Mg - N$ and $- ve$ due to the normal from the wall.
Finally the rod will come to rest on the ground so we can see that it has moved down and in left direction.

AIIMS-2005

Rotational Motion

149735 Two particles of mass $1 kg$ and $3 kg$ have position vectors $(2 \hat{i} + 3 \hat{j} + 4 \hat{k})$ and $(- 2 \hat{i} + 3 \hat{j} - 4 \hat{k})$ respectively. The centre of mass, has a position vector

1

\hat{i} + 3 \hat{j} - 2 \hat{k}

2

- \hat{i} - 3 \hat{j} - 2 \hat{k}

3

- \hat{i} + 3 \hat{j} + 2 \hat{k}

4

- \hat{i} + 3 \hat{j} - 2 \hat{k}

Explanation:

D Let $m_{1} = 1 kg, m_{2} = 3 kg$
${\vec{r}}_{1} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} and {\vec{r}}_{2} = - 2 \hat{i} + 3 \hat{j} - 4 \hat{k}$
$∴ {\vec{r}}_{CM} = \frac{m_{1} {\vec{r}}_{1} + m_{2} {\vec{r}}_{2}}{m_{1} + m_{2}}$
$= \frac{1 (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) + 3 (- 2 \hat{i} + 3 \hat{j} - 4 \hat{k})}{1 + 3}$
${\vec{r}}_{CM} = - \hat{i} + 3 \hat{j} - 2 \hat{k}$

EAMCET-1999

Rotational Motion

149736 The velocities of three particles of masses $20 g$ , $30 g$ and $50 g$ are $10 \hat{i}, 10 \hat{j}$ and $10 \hat{k}$ respectively. The velocity of the centre of mass of the three particles is

1

2 \hat{i} + 3 \hat{j} + 5 \hat{k}

2

10 (\hat{i} + \hat{j} + \hat{k})

3

20 \hat{i} + 30 \hat{j} + 5 \hat{k}

4

2 \hat{i} + 30 \hat{j} + 50 \hat{k}

Explanation:

A Let $m_{1}, m_{2}, m_{3}$ be $20 g, 30 g, 50 g$ respectively and ${\vec{V}}_{1}, {\vec{V}}_{2}, {\vec{V}}_{3}$ be $10 \hat{i}, 10 \hat{j}, 10 \hat{k}$ respectively
Then, ${\vec{V}}_{CM} = \frac{m_{1} {\vec{V}}_{1} + m_{2} {\vec{V}}_{2} + m_{3} {\vec{V}}_{3}}{m_{1} + m_{2} + m_{3}}$
$= \frac{20 \times 10 \hat{i} + 30 \times 10 \hat{j} + 50 \times 10 \hat{k}}{20 + 30 + 50}$
$= \frac{200 \hat{i} + 300 \hat{j} + 500 \hat{k}}{100}$
${\vec{V}}_{C M} = 2 \hat{i} + 3 \hat{j} + 5 \hat{k}$

EAMCET-2001